Unimodular toric ideals of graphs

TL;DR

This work classifies graphs whose toric ideals are unimodular by establishing a complete graph-theoretic criterion: $I_G$ is unimodular if and only if $G$ has the strong odd cycle property, i.e., any two odd cycles intersect. The authors leverage the toric bases (circuits, Graver basis, universal Gröbner basis) and known circuit/primitive-walk characterizations to connect unimodularity with concrete graph structures, including the concepts of link vertices and flower-graphs. They provide a structural, constructive description—via $H$-paths and parity constraints—for building all unimodular graphs, covering both bipartite and non-bipartite cases and presenting a complete algorithmic classification. This advances understanding of when incidence matrices yield unimodular toric ideals, with implications for algebraic statistics, combinatorial commutative algebra, and related computational applications.

Abstract

We give a necessary and sufficient graph-theoretic characterization of toric ideals of graphs that are unimodular. As a direct consequence, we provide the structure of unimodular graphs by proving that the incidence matrix of a graph $G$ is unimodular if and only if any two odd cycles of $G$ intersect.

Figures (5)

• Figure 1: The cases that circuits and primitive elements coincide$;$$c$ is an even cycle, $v$ a vertex, $c_1,c_2$ are odd cycles and $p$ a path
• Figure 2: The case that a primitive element is not a circuit$;$$c_1,c_2$ are odd cycles which are joined by two disjoint paths $p_1$ and $p_2$ of the same parity$;$ i.e. they are both even or odd
• Figure 3: A not unimodular graph. All its blocks are unimodular.
• Figure 4: Unimodular graphs. The graph on the left consists of one non bipartite block with the strong odd cycle property. The graph on the right consists of four odd cycles with one link vertex $v$.
• Figure 5: The $H$-paths of Theorem \ref{['construction!']}. The first two figures correspond to the case ($\alpha$), the third figure correspond to the case ($\beta$), and the last one correspond to the case ($\gamma$)

Theorems & Definitions (31)

• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• Example 2.4
• Theorem 2.5
• Proposition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4